2 - Calculus of Variations
from PART I - VARIATIONAL METHODS
Published online by Cambridge University Press: 05 July 2013
Summary
To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature … If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in.
(Richard Feynman)Calculus is the mathematics of change. Differential calculus addresses the change of a function as one moves from point to point. A function u(x) is such that the dependent variable u takes on a unique value for each value of the independent variable x, and the derivative of the function at a point indicates the rate of change, or slope, at that point. In calculus of variations, we deal with the functional, which may be regarded as a “function of functions,” that is, a function that depends on other functions. More specifically, a functional is a definite integral whose integrand contains a function that is yet to be determined. A functional I[u(x)] is such that I takes on a unique scalar value for each function u(x). In Section 1.3, the travel time T[u(x)] and total energy E[u(z)] are functionals, which are functions of the path of light or the boat u(x) and bubble shape u(z), respectively. Variational calculus addresses the change in a functional as one moves from function to function. Accordingly, whereas differential calculus is the calculus of functions, variational calculus is the calculus of functionals.
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- Publisher: Cambridge University PressPrint publication year: 2013